Method of on-line monitoring of radial clearances in steam turbines

ABSTRACT

A method of monitoring radial clearances in a steam turbine during operation of the turbine is provided. The method, in an exemplary embodiment, includes measuring a temperature of the rotor shaft at a time 1  and at a time 2 , measuring a temperature of the rotor blade at time 1  and at time 2 , measuring a temperature of the shell at time 1  and at time 2 , calculating a shaft radial growth between time 1  and time 2 , calculating a blade growth between time 1  and time 2 , calculating a shell radial growth between time 1  and time 2 , and determining a change in a radial gap between the shell and a distal end of the rotor blade from time 1  to time 2  using the following equation: change in radial gap=shell radial growth−shaft radial growth−blade growth.

BACKGROUND OF THE INVENTION

[0001] The present invention relates generally to rotary machines, suchas steam and gas turbines, and, more particularly, relates to a methodof monitoring clearance between tips of rotating rotor blades and astationary outer casing of a reaction design high pressure steamturbine.

[0002] Steam and gas turbines are used, among other purposes, to powerelectric generators. A steam turbine has a steam path which typicallyincludes, in serial-flow relationship, a steam inlet, a turbine, and asteam outlet. A gas turbine has a gas path which typically includes, inserial-flow relationship, an air intake (or inlet), a compressor, acombustor, a turbine, and a gas outlet (or exhaust nozzle). Compressorand turbine sections include at least one circumferential row ofrotating blades. The free ends or tips of the rotating blades aresurrounded by a stator casing.

[0003] The efficiency of the turbine depends in part on the radialclearance or gap between the rotor blade tips and the surrounding casingand the clearance between the rotor and the diaphragm packings. If theclearance is too large, more of the steam or gas flow will leak throughthe gap between the rotor blade tips and the surrounding casing orbetween the diaphragm and the rotor, decreasing the turbine'sefficiency. If the clearance is too small, the rotor blade tips canstrike the surrounding casing during certain turbine operatingconditions. Gas or steam leakage, either out of the gas or steam path orinto the gas or steam path, from an area of higher pressure to an areaof lower pressure, is generally undesirable. For example, gas-pathleakage in the turbine or compressor area of a gas turbine, between therotor of the turbine or compressor and the circumferentially surroundingturbine or compressor casing, will lower the efficiency of the gasturbine leading to increased fuel costs. Also, steam-path leakage in theturbine area of a steam turbine, between the rotor of the turbine andthe circumferentially surrounding casing, will lower the efficiency ofthe steam turbine leading to increased fuel costs.

[0004] It is known that the clearance changes during periods ofacceleration or deceleration due to changing centrifugal force on theblade tips and due to relative thermal growth between the rotating rotorand stationary casing. During periods of differential centrifugal andthermal growth of the rotor and casing the clearance changes can resultin severe rubbing of the moving blade tips against the stationarycasing. This increase in blade tip clearance results in efficiency loss.

[0005] Clearance control devices, such as rigid abradable shrouds, havebeen used in the past to accommodate rotor-to-casing clearance change.However, none are believed to represent an optimum design forcontrolling such clearance. Also, positive pressure packings have beenused that include movable packings that permit the packings to be in aretracted position during startup and in an extended position duringsteady state operation of the turbine. However, the moving parts canstick during operation preventing the packings from moving between theextended and retracted positions.

BRIEF DESCRIPTION OF THE INVENTION

[0006] In one aspect, a method of monitoring radial clearances in asteam turbine during operation of the turbine is provided. The turbineincludes an outer shell and a rotor including a rotor shaft and aplurality of rotor blades attached to the rotor shaft. The methodincludes measuring a temperature of the rotor shaft at a time₁ and at atime₂, measuring a temperature of the rotor blade at time₁ and at time₂,measuring a temperature of the shell at time₁ and at time₂, calculatinga shaft radial growth between time₁ and time₂, calculating a bladegrowth between time₁ and time₂, calculating a shell radial growthbetween time₁ and time₂, and determining a change in a radial gapbetween the shell and a distal end of the rotor blade from time₁ totime₂ using the following equation: change in radial gap=shell radialgrowth−shaft radial growth−blade growth.

[0007] In another aspect, a method of monitoring radial clearances in asteam turbine during operation of the turbine is provided. The turbineincludes an outer shell and a rotor including a rotor shaft and aplurality of rotor blades attached to the rotor shaft. The methodincludes measuring a temperature of the rotor shaft continuously duringoperation, measuring a temperature of the rotor blade continuouslyduring operation, measuring a temperature of the shell continuouslyduring operation, calculating a shaft radial growth as a function ofrotor shaft temperature over time, calculating a blade growth as afunction of rotor blade temperature over time, calculating a shellradial growth as a function of shell temperature over time, anddetermining a change in a radial gap between the shell and a distal endof the rotor blade over time using the following equation: change inradial gap=shell radial growth−shaft radial growth−blade growth.

[0008] In another aspect, a method of monitoring radial clearances in asteam turbine during operation of the turbine is provided. The turbineincludes an outer shell and a rotor including a rotor shaft and aplurality of rotor blades attached to the rotor shaft. The methodincludes calculating a shaft radial growth as a function of rotor shafttemperature over time, calculating a blade growth as a function of rotorblade temperature over time, calculating a shell radial growth as afunction of shell temperature over time, and determining a change in aradial gap between the shell and a distal end of the rotor blade overtime using the following equation: change in radial gap=shell radialgrowth−shaft radial growth−blade growth.

BRIEF DESCRIPTION OF THE DRAWINGS

[0009]FIG. 1 is a sectional schematic representation of a reactiondesign steam turbine.

[0010]FIG. 2 is an enlarged sectional schematic representation of aportion of the steam turbine shown in FIG. 1.

[0011]FIG. 3 is a flow chart of a method of monitoring radial clearancesin a steam turbine during operation of the turbine.

[0012]FIG. 4 is a schematic representation of a portion of a steamturbine rotor.

DETAILED DESCRIPTION OF THE INVENTION

[0013] A method of monitoring radial clearances in a steam turbineduring operation of the turbine is described in more detail below. Themethod calculates thermal expansions of components in the steam turbinewhich are proportional to averaged metal temperatures at a givenlocation in the turbine. For example, the averaged temperature for theturbine shell at a given location can be obtained from measurements ofshell temperature at one or more points across the thickness of theshell. Also, the temperature distribution in the rotor at a givenlocation can be computed from the measured surface temperature and therate of change of surface temperature over time. The method is describedas used in a reaction design steam turbine; however, the methoddescribed below is applicable for other steam turbine designs, such asimpulse steam turbines. The method uses the measured data from turbineshells and rotors for on-line computations of radial clearances ofturbine components. This real time clearance data can be used by anoperator to control turbine transients such that tip clearance changesare within specified limits for high thermal efficiency and to avoidrubbing between rotor tips and the shell.

[0014] Referring to the drawings, FIG. 1 is a sectionalschematic-representation of a reaction design steam turbine 10. Steamturbine 10 includes a rotor shaft 12 passing through turbine 10 andsealed at each end by packings 14. A plurality of turbine blades 16 areconnected to shaft 12. Between turbine blades 16 there is positioned aplurality of non-rotating turbine nozzles 18. Turbine blades or buckets16 are connected to turbine shaft 12 while turbine nozzles 18 extendfrom an inner housing or shell 20 surrounding turbine blades 16 andnozzles 18. An outer housing 22 encloses inner housing 20 and rotorshaft 12 Steam is directed through nozzles 18 and through blades 16causing blades 16 to rotate along with turbine shaft 12.

[0015]FIG. 2 is an enlarged sectional schematic representation of innerhousing 20 of steam turbine 10. Inner housing 20 includes a plurality ofouter ring portions 30. Each outer ring portions 30 include a ring 32 ofsteam directing nozzles 18 supported within outer ring portion 30, andan inner ring portion 34 contained within nozzle ring 32. Turbinebuckets 16 are secured at their inner ends 36 to turbine wheels 38extending from turbine shaft 12 rotatable about an axis 40. The radialouter ends 42 of buckets 16 include bucket covers 44 which rotate withbuckets 16. In one embodiment, a cover 44 is positioned on radial outerend 42 of each bucket 16 and in alternate embodiments on outer ends 42of two or more buckets 16 in the form of a band so as to permit adjacentbuckets 16 to be coupled to a common cover or band.

[0016] Inner ring portion 34 of housing 20 includes a packing ring 48.Packing ring 48 is positioned adjacent turbine shaft 12. Turbine shaft12 includes a sealing means 54 to seal a gap 56 between turbine shaft 12and inner ring portion 34 of inner housing 20 to prevent the passage ofstream through gap 56. Sealing means 54 is positioned adjacent packingring 48 and includes a plurality of axially spaced brush seals 58extending from rotor 12. Sealing means 54 can also include axiallyspaced labyrinth seal teeth (not shown) or a combination of axiallyspaced labyrinth seal teeth and brush seal seals 58.

[0017] Bucket covers 44 include a sealing means 66 to provide a seal ina gap 68 between bucket cover 44 and housing 20 to prevent the passageof steam through gap 68. Sealing means 66 includes a plurality ofaxially spaced labyrinth seal teeth 70 extending from bucket cover 44.Sealing means 66, in other embodiments include brush seals alone orcombined with axially spaced labyrinth seal teeth 70.

[0018] Referring to FIG. 3, a method 80 of monitoring radial clearances,for example the size of gap 68 and the size of gap 56, in steam turbine10 includes measuring 82 a temperature of rotor shaft 12 a time₁ and ata time₂, measuring 84 a temperature of rotor blade 16 time₁ and attime₂, measuring 86 a temperature of shell 20 at time₁ and at time₂,calculating 88 a radial growth of shaft 12 between time₁ and time₂,calculating 90 a growth of blade 16 between time₁ and time₂, calculating92 a radial growth of shell 20 between time₁ and time₂, and determining94 a change in radial gap 68 from time₁ to time₂ using the equation:

Change In Radial Gap=Shell Radial Growth−Shaft Radial Growth−BladeGrowth.

[0019] Of course, because the radial growth of blade 16 has no effect ongap 56, the term Blade Growth in the above equation is zero forcalculations of the change in radial growth of gap 56. Also, thetemperatures of shaft 12, shell 20 and blade 16 can be measured atdistinct intervals or can be continuously monitored over time.

[0020] The shaft radial growth is equal to the coefficient of thermalexpansion of the rotor (α_(R)) times an outer radius (R_(R)) of therotor times an instantaneous volume averaged temperature (T_(R)) of therotor.

Shaft Radial Growth=α_(R) *R _(R) *T _(R)

[0021] The blade radial growth is equal to the coefficient of thermalexpansion of the rotor blade (α_(B)) times a length (L_(B)) of the rotorblade times an instantaneous volume averaged temperature (T_(B)) of therotor blade.

Rotor Blade Growth=α_(B) *L _(B) *T _(B)

[0022] In most cases the instantaneous volume averaged temperature(T_(B)) of the rotor blade can be closely approximated by the rotorouter surface temperature or the steam temperature.

[0023] The shell radial growth is equal to the coefficient of thermalexpansion of the shell (α_(S)) times an inner radius (R_(S)) of theshell at the blade tip times an instantaneous volume averagedtemperature (T_(S)) of the shell.

Shell Radial Growth=α_(S) *R _(S) *T _(S)

[0024] In double wall shell designs with horizontal flanges, the radialclearance can vary as a function of circumferential location on theshell. To account for these variances shell radial growth is calculatedat the top, the bottom and the side of the shell. Particualrly, theinstantaneous volume averaged temperature (T_(S)) of the shell iscalculated for each location, at the top, the bottom and the side of theshell.

[0025] Instantaneous average temperatures T_(R) and T_(S) are computedusing a finite difference method employing a finite element modelutilizing the finite element of a segment of an infinitely longcylinder. This method is explained below using the rotor as an example,and the same method is applicable for the shell, considering the shellas a hollow cylinder. Referring to FIG. 4, the rotor is divided in aspecific number of elements, for example 10 elements.

[0026] Controls

[0027] Elements=10 (Elements number)

[0028] Nodes=Elements+2 (Nodes number)

[0029] Nr=Nodes−1 (Last Centroid Node Number)${Volume} = {\frac{\left( {R_{o}^{2} - R_{i}^{2}} \right)}{Elements}\left\lbrack {in}^{3} \right\rbrack}$

[0030] Volume→Element Volume

[0031] The Temperature & Time Maximum Incremental Changes are set.

[0032] MaxDTemp=5 (Maximum Incremental Temperature Change)${{Max}\quad {DTime}} = {\frac{\left( {R_{o} - R_{i}} \right)^{2}}{8 \cdot {DO} \cdot {Elements}^{2}}\quad\left\lbrack \min \right\rbrack}$

[0033] MaxDTime→Maximum Incremental Time Change

[0034] Initializing Temperature

[0035] This block assigns an initial temperature value to each boundaryof the rotor elements.

[0036] For I=1 To Nodes

[0037] Tr(I)=InitialTemp [° F.]

[0038] Next I

[0039] Tr(Nodes)→Array of Nodes elements.

[0040] Since Tr(Nodes) represents the boundary temperatures of eachrotor element, all Tr elements have an initial value of InitialTemp.

[0041] Initial Temperature Distribution

[0042] Tsurf=InitialTemp: Tavg=InitialTemp: Tbore=InitialTemp

[0043] Since the initial temperature distribution of the rotor isuniform, the rotor surface temperature (Tsurf), the rotor boretemperature (Tbore) and the rotor average temperature (Tavg) have aninitial value of InitialTemp.

[0044] Centrifugal Stress Factor

[0045] Due to the large stress gradient existing at the rotor bore, theBoreCntrfStrs (Bore Centrifugal Stress) needs to be evaluated. Thisblock calculates the Centrifugal Stress Factor (SpeedFact) needed tocalculate the Bore Centrifugal Stress defined as:${SpeedFact} = {\frac{InBoreCntrStrs}{{RatedSpeed}^{\quad 2}}\left\lbrack {{KSI}/{RPM2}} \right\rbrack}$

[0046] Extrapolation Factor

[0047] To calculate the thermal stress at the surfaces at any giventime, Tsurf, Tbore, and Tavg must be known. The extrapolation factor(Extrapfact) is found by means of an extrapolation of the ramp ratetemperatures to the inner surface.

[0048] If we consider that Ri always has a greater value than 0, ispossible to eliminate the second option that considers the case forRi=0. $R = {\sqrt{R_{i}^{2} + \frac{Volume}{2}}\lbrack{in}\rbrack}$${R2} = {\sqrt{R^{2} + \frac{Volume}{2}}\lbrack{in}\rbrack}$${Extrapfact} = {\left( \frac{R^{2} - {R_{i}^{2} \cdot \left( {1 + {2 \cdot {\log \left( \frac{R}{R_{i}} \right)}}} \right)}}{{Volume} - {2 \cdot R_{i}^{2} \cdot {\log \left( \frac{R_{2}}{R} \right)}}} \right)\lbrack{Dimensionless}\rbrack}$

[0049] ExtrapFact→Extrapolation Factor for Temperature at Bore Surface

[0050] Heatflow Factors

[0051] This block assigns a heatflow factor for the internal elements ofthe rotor. The heatflow factor for a specific rotor element is theaverage area normal to qi divided by the distance from element i toelement i+1.

For I=2 To Nr−1

R2=Sqr(R{circumflex over ( )}2+Volume) [in]

A(I)=(R+R2)/(R2−R) [in]

R=R2

Next I

[0052] Since the Finite Difference Method uses the conservation ofenergy for a specific rotor element to get the radial temperaturedistribution, the values of the heat flow factors vary radially too.

[0053] Surface & Bore Heatflow Factors

[0054] The previous formula to calculate the heatflow factors appliesdirectly to all internal elements, but must be modified for the boundaryelements to meet the boundary conditions. This block assigns theheatflow factor for the boundary elements.

A(1)=0 [in]

A(Nr)=(R+Ro)/(Ro−R) [in]

Asurf=2*Ro/A(Nr) [in]

A(Nr)→Array of Nr elements.

[0055] Ramp Rates

[0056] This block defines the Time, Temperature, HTC and Speed RampRates using the inputs of the previous block. The variable NumSubStepsdefines the number of iterations of the new temperature distributionblock.

DTime=Time−OldTime [min]

DTemp=Tfluid−OldTfluid [° F.]

TempRamp=DTemp/DTime [° F./min]

HTCRamp=(HTC−OldHTC)/DTime [BTU/hr·ft2·° F./min]

SpeedRamp=(Speed−OldSpeed)/DTime [RPM/min]

NumSubSteps=Int(Application.WorksheetFunction_(—)

.Max(DTime/MaxDTime, Abs(DTemp/MaxDTemp), 1)) [Dimensionless]

DT=DTime/NumSubSteps [min]

[0057] The accuracy of the calculation of the new temperaturedistribution calculation depends on the size of the time step.Sufficient accuracy is obtained if the maximum time step is DT.

[0058] New Temperature Distribution

[0059] This block calculates the new temperature distribution of therotor setting new values for Tr elements.

For K=1 To NumSubSteps

TrSum=0

For I=2 To Nr

DF=Diff(D0, DM, Tr(I)) [in2/min]

TrNew=Tr(I)−DT*DF/Volume*(A(I)*(Tr(I)−Tr(I+1))+A(I−1)*(Tr(I)−Tr(I−1)))[° F.]

TrSum=TrSum+TrNew [° F.]

Tr(I−1)=PrevNew [° F.]

PrevNew=TrNew [° F.]

Next I

[0060] The variable TrNew calculates the new temperature value for aspecific rotor element using the corresponding values of thermaldiffusivity and heatflow factor. The variable TrSum allows the requiredstorage of information to calculate the average temperature (Tavg).

[0061] Time Delta

[0062] This block assigns new values for Time, Temperature and Speedusing the maximum time step (DT).

Time=OldTime+K*DT [min]

Tfluid=OldTfluid+TempRamp*K*DT [° F.]

Speed=OldSpeed+SpeedRamp*K*DT [RPM]

Tr(Nr)=TrNew [° F.]

[0063] Surface Temperature & HTC

[0064] This block assigns a new value for HTC and calculates the surfacetemperature (Tsurf). Since a convection heat transfer process is carriedout between the rotor surface and the fluid, the fluid temperature(Tfluid) is required to calculate the temperature at the last rotorelement Tr(Nodes).

HTC=OldHTC+HTCRamp*K*DT [BTU/hr·ft2·° F.]

Cond=RhoC*Diff(D0, DM, Tr(Nodes)) [BTU·° F./in·min]   (A)

Factor=(HTC/8640)*Asurf/Cond [Dimensionless]

Tr(Nodes)=(Tr(Nr)+Factor*Tfluid)/(1+Factor) [° F.]

Tsurf=Tr(Nodes) [° F.]

[0065] Bore & Average Temperatures

[0066] This block calculates the bore temperature (Tbore) and theaverage temperature (Tavg). To calculate the temperature of the firstrotor element the extrapolation factor for temperature at bore surface(ExtrapFact) is required.

Tr(1)=Tr(2)−ExtrapFact*(Tr(3)−Tr(2)) [° F.]

Tbore=Tr(1) [° F.]

Tavg=TrSum/Elements [° F.]

[0067] Surface Stress & Strain

[0068] This block calculates the surface stress and strain. The actualcoefficient of thermal expansion (Alpha) is required.

ExpnC=Alpha(A0, AM, Tavg) [%/° F.]

SurfStrn=ExpnC*(Tavg−Tsurf) [%]

SurfStrs=Modulus(E0, EM, Tsurf)*SurfStm/0.7 [KSI]

PCSurfAllow=100*SurfStrn/AllowSurfStrn [%]

[0069] Bore Stress

[0070] This block calculates the total bore stress. The actualcoefficient of thermal expansion (Alpha) and the actual Young's modulus(Modulus) are required.

BoreCntrfStrs=SpeedFact*Speed{circumflex over ( )}2 [KSI]

BoreThrmStrs=Modulus(E0, EM, Tbore)*ExpnC*(Tavg−Tbore)/0.7 [KSI]

TotBoreStrs=BoreThrmStrs+BoreCntrfStrs [KSI]

PCBoreAllow=100*TotBoreStrs/AllowBoreStrs [%]

Next K

[0071] While the invention has been described in terms of variousspecific embodiments, those skilled in the art will recognize that theinvention can be practiced with modification within the spirit and scopeof the claims.

1. A method of monitoring radial clearances in a steam turbine duringoperation of the turbine, the turbine comprising an outer shell and arotor, the rotor comprising a rotor shaft and a plurality of rotorblades attached to the rotor shaft, said method comprising: measuring atemperature of the rotor shaft at a time₁ and at a time₂; measuring atemperature of the rotor blade at time₁ and at time₂; measuring atemperature of the shell at time₁ and at time₂; calculating a shaftradial growth between time₁ and time₂; calculating a blade growthbetween time₁ and time₂; calculating a shell radial growth between time₁and time₂; and determining a change in a radial gap between the shelland a distal end of the rotor blade from time₁ to time₂ using thefollowing equation: change in radial gap=shell radial growth−shaftradial growth−blade growth.
 2. A method in accordance with claim 1wherein calculating a shaft radial growth comprises calculating a shaftradial growth using the following equation: shaft radial growth=α _(R)*R _(R)*T_(R) where α_(R) is the coefficient of thermal expansion of therotor; R_(R) is an outer radius of the rotor; T_(R) is an instantaneousvolume averaged temperature of the rotor:
 3. A method in accordance withclaim 1 wherein calculating a rotor blade growth comprises calculating arotor blade growth using the following equation: rotor bladegrowth=α_(B) *L _(B) *T _(B) where α_(B) is the coefficient of thermalexpansion of the blade; L_(B) is a length of the blade; T_(B) is aninstantaneous volume averaged temperature of the blade.
 4. A method inaccordance with claim 1 wherein calculating a shell radial growthcomprises calculating a shell radial growth using the followingequation: shell radial growth=α_(S) *R _(S) *T _(S) where α_(S) is thecoefficient of thermal expansion of the shell; R_(S) is an inner radiusof the shell at the blade tip; T_(S) is an instantaneous volume averagedtemperature of the shell.
 5. A method in accordance with claim 4 whereinT_(S) is an instantaneous volume averaged temperature of the shell at atop location.
 6. A method in accordance with claim 4 wherein T_(S) is aninstantaneous volume averaged temperature of the shell at a bottomlocation.
 7. A method in accordance with claim 4 wherein T_(S) is aninstantaneous volume averaged temperature of the shell at a sidelocation.
 8. A method of monitoring radial clearances in a steam turbineduring operation of the turbine, the turbine comprising an outer shelland a rotor, the rotor comprising a rotor shaft and a plurality of rotorblades attached to the rotor shaft, said method comprising: measuring atemperature of the rotor shaft continuously during operation; measuringa temperature of the rotor blade continuously during operation;measuring a temperature of the shell continuously during operation;calculating a shaft radial growth as a function of rotor shafttemperature over time; calculating a blade growth as a function of rotorblade temperature over time; calculating a shell radial growth as afunction of shell temperature over time; and determining a change in aradial gap between the shell and a distal end of the rotor blade overtime using the following equation: change in radial gap=shell radialgrowth−shaft radial growth−blade growth.
 9. A method in accordance withclaim 8 wherein calculating a shaft radial growth comprises calculatinga shaft radial growth using the following equation: shaft radialgrowth=α_(R) *R _(R) *T _(R) where α_(R) is the coefficient of thermalexpansion of the rotor; R_(R) is an outer radius of the rotor; T_(R) isan instantaneous volume averaged temperature of the rotor.
 10. A methodin accordance with claim 8 wherein calculating a rotor blade growthcomprises calculating a rotor blade growth using the following equation:rotor blade growth=α_(B) *L _(B) *T _(B) where α_(B) is the coefficientof thermal expansion of the blade; L_(B) is a length of the blade; T_(B)is an instantaneous volume averaged temperature of the blade.
 11. Amethod in accordance with claim 8 wherein calculating a shell radialgrowth comprises calculating a shell radial growth using the followingequation: shell radial growth=α_(S) *R _(S) *T _(S) where α_(S) is thecoefficient of thermal expansion of the shell; R_(S) is an inner radiusof the shell at the blade tip; T_(S) is an instantaneous volume averagedtemperature of the shell.
 12. A method in accordance with claim 11wherein T_(S) is an instantaneous volume averaged temperature of theshell at a top location.
 13. A method in accordance with claim 11wherein T_(S) is an instantaneous volume averaged temperature of theshell at a bottom location.
 14. A method in accordance with claim 11wherein T_(S) is an instantaneous volume averaged temperature of theshell at a side location.
 15. A method of monitoring radial clearancesin a steam turbine during operation of the turbine, the turbinecomprising an outer shell and a rotor, the rotor comprising a rotorshaft and a plurality of rotor blades attached to the rotor shaft, saidmethod comprising: calculating a shaft radial growth as a function ofrotor shaft temperature over time; calculating a blade growth as afunction of rotor blade temperature over time; calculating a shellradial growth as a function of shell temperature over time; anddetermining a change in a radial gap between the shell and a distal endof the rotor blade over time using the following equation: change inradial gap=shell radial growth−shaft radial growth−blade growth.
 16. Amethod in accordance with claim 15 wherein calculating a shaft radialgrowth comprises calculating a shaft radial growth using the followingequation: shaft radial growth=α_(R) *R _(R) *T _(R) where α_(R) is thecoefficient of thermal expansion of the rotor; R_(R) is an outer radiusof the rotor; T_(R) is an instantaneous volume averaged temperature ofthe rotor.
 17. A method in accordance with claim 15 wherein calculatinga rotor blade growth comprises calculating a rotor blade growth usingthe following equation: rotor blade growth=α_(B) *L _(B) *T _(B) whereα_(B) is the coefficient of thermal expansion of the blade; L_(B) is alength of the blade; T_(B) is an instantaneous volume averagedtemperature of the blade.
 18. A method in accordance with claim 15wherein calculating a shell radial growth comprises calculating a shellradial growth using the following equation: shell radial growth=α_(S) *R_(S) *T _(S) where α_(S) is the coefficient of thermal expansion of theshell; R_(S) is an inner radius of the shell at the blade tip; T_(S) isan instantaneous volume averaged temperature of the shell.
 19. A methodin accordance with claim 18 wherein T_(S) is an instantaneous volumeaveraged temperature of the shell at a top location.
 20. A method inaccordance with claim 18 wherein T_(S) is an instantaneous volumeaveraged temperature of the shell at a bottom location.
 21. A method inaccordance with claim 18 wherein T_(S) is an instantaneous volumeaveraged temperature of the shell at a side location.
 22. A system formonitoring radial clearances in a steam turbine during operation of theturbine, the turbine comprising an outer shell and a rotor, the rotorcomprising a rotor shaft and a plurality of rotor blades attached to therotor shaft, said system comprising: a measurement means configured to:measure a temperature of the rotor shaft at a time₁ and at a time₂;measure a temperature of the rotor blade at time₁ and at time₂; measurea temperature of the shell at time₁ and at time₂; and a calculationmeans configured to: calculate a shaft radial growth between time₁ andtime₂; calculate a blade growth between time₁ and time₂; calculate ashell radial growth between time₁ and time₂; and calculate a change in aradial gap between the shell and a distal end of the rotor blade fromtime₁ to time₂ using the following equation: change in radial gap=shellradial growth−shaft radial growth−blade growth.
 23. A method inaccordance with claim 22 wherein said calculation means is furtherconfigured to calculate the shaft radial growth using the followingequation: shaft radial growth=α_(R) *R _(R) *T _(R) where α_(R) is thecoefficient of thermal expansion of the rotor; R_(R) is an outer radiusof the rotor; T_(R) is an instantaneous volume averaged temperature ofthe rotor.
 24. A system in accordance with claim 22 wherein saidcalculation means is further configured to calculate the rotor bladegrowth using the following equation: rotor blade growth=α_(B) *L _(B) *T_(B) where α_(B) is the coefficient of thermal expansion of the blade;L_(B) is a length of the blade; T_(B) is an instantaneous volumeaveraged temperature of the blade.
 25. A system in accordance with claim22 wherein said calculation means is further configured to calculate theshell radial growth using the following equation: shell radialgrowth=α_(S) *R _(S) *T _(S) where α_(S) is the coefficient of thermalexpansion of the shell; R_(S) is an inner radius of the shell at theblade tip; T_(S) is an instantaneous volume averaged temperature of theshell.
 26. A system in accordance with claim 25 wherein T_(S) is aninstantaneous volume averaged temperature of the shell at a toplocation.
 27. A system in accordance with claim 25 wherein T_(S) is aninstantaneous volume averaged temperature of the shell at a bottomlocation.
 28. A system in accordance with claim 25 wherein T_(S) is aninstantaneous volume averaged temperature of the shell at a sidelocation.